Games and Economic Behavior 82 (2013) 503–521

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Games and Economic Behavior www.elsevier.com/locate/geb

Asymmetric awareness and moral hazard Sarah Auster Department of Economics, European University Institute, Via dei Roccettini 9, I-50014 San Domenico di Fiesole (FI), Italy

a r t i c l e

i n f o

Article history: Received 8 June 2012 Available online 10 September 2013 JEL classification: D01 D83 D86 Keywords: Unawareness Moral hazard Incomplete contracts

a b s t r a c t This paper introduces asymmetric awareness into the classical principal–agent model and discusses the optimal contract between a fully aware principal and an unaware agent. The principal enlarges the agent’s awareness strategically when proposing a contract and faces a tradeoff between participation and incentives. Leaving the agent unaware allows the principal to exploit the agent’s incomplete understanding of the world, relaxing the participation constraint, while making the agent aware enables the principal to use the revealed contingencies as signals about the agent’s action choice, relaxing the incentive constraint. The optimal contract reveals contingencies that have low probability but are highly informative about the agent’s effort. © 2013 Elsevier Inc. All rights reserved.

1. Introduction The canonical moral hazard model analyzes the optimal contract between a principal and an agent in the presence of privately observable effort. As in most economic models, the underlying assumption is that principal and agent are fully aware of every possible outcome realization and its distributional properties. However, in reality there are contracting situations where one party has a better understanding of the underlying uncertainties than the other. The question this paper addresses is whether the party with superior awareness can use his better understanding of the world strategically in the presence of moral hazard. To illustrate this, consider the owner of a firm who wants to hire a manager. It is possible that the firm owner is aware of more opportunities and liabilities concerning his firm than the manager. Suppose, for example, that there is the possibility that one of the firm’s products has adverse effects on the health of consumers. As a consequence, it is possible that the firm has to recall the product and faces severe legal liabilities. Whether the product’s potential health threat becomes public or not is uncertain and depends on the manager’s effort. Suppose further that the possibility of this event never crossed the manager’s mind. The question is then, under which conditions it is profitable for the firm owner to disclose a possible recall and its legal consequences to the manager when offering the contract. The model shows that there is a trade off between participation and incentives. First, if the firm owner does not reveal the potential threat to the manager, he designs the contract such that the manager receives the minimal payment if the recall is realized. Since the manager does not take into account such an event, the participation constraint is less costly to satisfy. The size of the participation effect depends on the probability of the event. If the potential recall and its legal consequences are highly probable, it is easier to hire the manager without disclosing the possibility of the product’s health threat. Second, since the probability of the health threat becoming public depends on the effort of the manager, its realization is a signal about the manager’s effort. Disclosing the potential threat makes the incentive constraint less costly to satisfy. The size of the incentive effect depends on how much

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the manager’s effort affects the probability of the event. If the manager can reduce the probability of the potential recall significantly, it is optimal to make his payment contingent on its realization. This paper proposes a theoretical model which introduces asymmetric awareness in the canonical moral hazard model. The model analyzes the optimal contract between a fully aware principal and an unaware agent. A decision maker is called unaware when there exist contingencies that he does not know, and he does not know that he does not know, and so on ad infinitum (Modica and Rustichini, 1994, 1999). In the proposed model, the agent is assumed to be unaware of some relevant events, meaning that there are contingencies that affect the agent’s payoff but that have never crossed his mind. Further, the agent is assumed to be unaware of his unawareness, so he believes that his description of the world is correct and complete. This implies that the agent is oblivious to the possibility that the principal is aware of contingencies that he is unaware of. The principal, on the other hand, is assumed to be fully aware. Moreover, the principal knows that the agent is unaware and he knows what the agent is unaware of. When writing the contract the principal can make the agent aware of some or all relevant contingencies. Note that a contract which transmits awareness is distinct from a contract that transmits information. A contract carrying information generally narrows the state space of the agent, whereas a contract carrying awareness expands the agent’s state space by adding new dimensions. Thus, the analysis of the optimization problem of a principal with superior awareness complements the literature on moral hazard with an informed principal (e.g. Benabou and Tirole, 2003). The risk-neutral principal proposes a contract to the risk-averse agent. The principal is the owner of a risky project, whose outcome is a function of the realization of a finite number of elementary contingencies of which the agent only knows a subset. These contingencies can be thought of as elementary propositions that can be either true or false. The probability of a contingency to be realized depends on the agent’s privately taken effort. The agent’s effort can be high or low and it is assumed that implementing high effort is always optimal. Since the principal cannot observe the agent’s action, the terms of the contract have to be such that it is in the agent’s best interest to exert the level of effort the principal wishes to implement. The compensation scheme is made contingent on the observable and verifiable outcome, rewarding the agent for outcome realizations that are relatively likely under high effort. The agent is assumed to have limited liability, thus transfers have to be non-negative in each state of the world. If the principal leaves the agent unaware of some contingencies, there is a non-empty set of possible outcomes that the agent does not take into account. It is optimal for the principal to construct the contract such that the agent receives the minimal payment whenever an unforeseen outcome is realized. The main question this paper addresses is whether and under which conditions the principal enlarges the agent’s awareness. The rationale for leaving the agent unaware is what I refer to as the participation effect. If the agent is unaware, his beliefs are systematically biased, which is exploited by the principal. The principal pays in expectation less than the agent’s reservation utility, because there is positive probability that he pays zero and because the agent does not take this into account. The rationale for making the agent aware is what I refer to as the incentive effect. Since the probability of a contingency to be realized depends on the effort of the agent, including it in the contract allows the principal to use its realization as a signal about the agent’s action choice. This implies that the information structure is richer and providing incentives is less costly. The principal includes contingencies in the contract for which the incentive gains outweighs the participation loss, determined by the distributional properties of these contingencies. The participation cost of announcing a contingency is the payment to the agent in the states where the contingency is realized. This cost increases with the probability that the unforeseen contingency is realized. The gain of including the contingency is the richer information structure, where this gain increases with the informativeness of the signal. Roughly speaking, contingencies for which the incentive effect dominates the participation effect are contingencies that are very unlikely but highly informative. The characterization of the tradeoff between participation and incentive effect is the key contribution of this paper. If the agent is unaware after reading the contract, his perception of the world differs from the perception of the principal. The question arises whether the agent can rationalize the proposed contract given his beliefs or whether he should get suspicious. To answer this question I analyze the principal’s optimization problem from the viewpoint of the agent. The solution to this problem coincides with the proposed contract whenever the principal’s expected profit evaluated at the agent’s beliefs is non-negative and the optimal effort choice is the same for both beliefs. Given these conditions, the proposed contract is fully rationalizable for the agent, i.e. the agent has no reason to become suspicious upon reading the contract. The reason for this is that the principal uses the signals within the agent’s awareness optimally. Since the agent is unaware of the existence of other relevant contingencies, the proposed contract maximizes the principal’s expected payoff evaluated at the agent’s beliefs. Next, I allow for competition among principals. In the benchmark model without unawareness, principals engage in a Bertrand competition over the compensation scheme. In equilibrium, principals make zero profits and the second-best surplus goes to the agent. If the agent is unaware, the symmetric awareness equilibrium exists but there may be other equilibria in which the agent stays unaware, even when competition is tight. As these equilibria are generally inefficient, this result is rather surprising. The reason why unawareness may persist in equilibrium is that if the agent’s perception of the world is sufficiently distorted, principals can make “generous” offers for outcomes within the agent’s awareness. Revealing the unforeseen states allows the agent to adjust his action choice, exploiting the equilibrium offer in his favor at the cost of the principals. If the agent can reduce the probability to receive the minimal payment sufficiently, accepting the equilibrium contract may yield a higher payoff than the second-best surplus, making any deviating contract unprofitable.

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Finally, some generalizations are discussed. Throughout the main part of the analysis it is assumed that output is a discrete one-to-one mapping from contingencies to real numbers. I discuss how results change when more general forms of output functions are considered. Next, the analysis abstracts from the optimal action choice. I show that whenever the principal wishes to implement low effort, it is optimal to not reveal any contingencies, because there is no incentive effect. Further, I extend the analysis to an environment with heterogeneous awareness of agents and show that unawareness is preserved in equilibrium only if the extent of initial unawareness is large enough. Next, I discuss the optimal contract when the agent is the residual claimant. There is an additional effect on the participation constraint because the agent’s evaluation of the project generally depends on his level of awareness. It is favorable to the principal to disclose negative outcome shocks, because their revelation lowers the agent’s outside option. Lastly, a frequently raised concern is whether unawareness is observationally equivalent to full awareness with zero probability beliefs. I discuss in what sense my model can be interpreted as a standard principal–agent model with heterogeneous priors. Section 2 gives an overview of the related literature. Section 3 introduces the theoretical model. In Section 4 the optimal contract with observable effort is characterized as a benchmark. The main part of the paper, Section 5, is devoted to the analysis of the optimal contract with unobservable effort. Section 6 introduces competition among principals and Section 7 discusses generalizations of the basic model. Section 8 offers some concluding remarks. All proofs can be found in Appendix A. 2. Related literature It is not possible to incorporate non-trivial unawareness in the standard state space model. This has been shown in the seminal paper by Dekel et al. (1998). In response, Heifetz et al. (2006), Li (2009), Board and Chung (2011) and Galanis (2013) have proposed generalized state space models that do allow for non-trivial unawareness. My model adopts the generalized state space model introduced by Heifetz et al. (2006). Their unawareness structure consists of a lattice of state spaces, ordered according to their expressive power, where each state space captures a particular horizon of propositions. In a companion work Heifetz et al. (2013) introduce probabilistic beliefs to the model. For ease of exposition, my model foregoes the formal introduction of state spaces, projections among them and events. Appendix A.1 explains how the basic model is built on the unawareness structure proposed by Heifetz et al. (2006/2013). Filiz-Ozbay (2012) was one of the first to incorporate unawareness into contracting problems. She considers a contracting situation between a fully aware insurer and an unaware insuree. The key difference between my work and her paper is the presence of moral hazard and the assumption on beliefs. In Filiz-Ozbay (2012) there is no hidden action and consequently no incentive effect. Her set up restricts my framework to the case where the agent is the residual claimant and the revelation of new states involves a participation effect only. However, Filiz-Ozbay (2012) allows for a wider range of equilibrium beliefs. She assumes that the agent assigns arbitrary probability beliefs to newly revealed states with the restriction that the principal’s payoff evaluated at the agent’s beliefs is non-negative and that relative probability beliefs previously held are unchanged. Given this assumption, it is possible that the agent’s beliefs deviate stronger from objective probabilities when becoming aware than before. Due to the insurance motive and the wider range of equilibrium beliefs, the effect of revelation on the participation constraint in her environment is ambiguous. Depending on the effect on the participation constraint, disclosure can be profitable or not. Also Ozbay (2008) analyzes a setting where the decision maker is unaware of some events and a fully aware announcer strategically mentions contingencies before the decision maker takes an action. Both Filiz-Ozbay (2012) and Ozbay (2008) explore the possibility that the unaware agent is able to reason why the other agent proposed the observed contract. A second strand of literature analyzes contracting problems with unawareness of actions. In these models agents are aware of all of nature’s moves but are unaware of their own action space. Von Thadden and Zhao (2012a, 2012b) propose a moral hazard model with a fully aware principal and an unaware agent. At first glance this set up may seem similar to mine, however the underlying intuition and the results are very distinct. In contrast to my model, the agent in their model understands all relevant contingencies but is unaware of his action space. Von Thadden and Zhao (2012a) assume that if the principal leaves the agent unaware, the agent chooses a default action unconsciously, but assesses his expected utility with respect to such default action correctly. The principal decides whether to make the agent aware of his action space or whether to leave him unaware. In a standard moral hazard framework this can be interpreted as the decision whether to restrict the agent’s action choice to some sub-optimal level ex-ante or whether to leave the action choice to the agent’s discretion. Making the agent aware enlarges the agent’s action space and consequently relaxes the participation constraint. However, enlarging the agent’s action space adds further incentive constraints to the principal’s optimization problem. Consequently, the principal faces a trade off between participation and incentives, but the effects are reversed compared to my model. Their main result is, that it is optimal to leave the agent unaware whenever the default action is close enough to the first best effort level. They extend their analysis to the case where agents differ in their level of awareness and derive the optimal menu of contracts. Also Zhao (2008) considers a moral hazard problem with unawareness of actions and default actions. In his setup both the principal and the agent can be unaware of their action space. Finally, this work is related to the literature on moral hazard and heterogeneous priors. Santos-Pinto (2008) analyzes a principal–agent model with an agent that holds wrong beliefs about the impact of his effort and calls such biased beliefs self-image. He shows that if positive self-image and effort are complements, the impact of positive self-image is favorable

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to the principal.1 If unawareness in my model is interpreted as assigning probability zero to certain outcomes, the resulting distribution does not satisfy the imposed restrictions in Santos-Pinto (2008). Consequently his results do not apply in my framework.2 Also De la Rosa (2011) analyzes a moral hazard problem with overconfidence. 3. The model There is a principal and an agent. The principal is risk neutral and the agent is risk averse. The agent receives utility from monetary transfers C and disutility from effort e. I assume that the utility function is separable in money and effort: U (C , e ) = υ (C ) − e, where υ satisfies the Inada conditions. Effort can take two possible values e ∈ {e L , e H }, where eL < eH . The uncertainty of the environment is captured by a finite set of elementary contingencies, denoted by Θ . A contingency θ ∈ Θ is a random variable with realizations 0 and 1. It can be thought of as an elementary proposition that can be either true or false. The probability of θ = 1 depends on the effort of the agent. Throughout the main part of the analysis it will be assumed that, given e, the contingencies in Θ are conditionally independent of each other. Assumption 1. The random variables θ and θ  are conditionally independent given e, for any θ , θ  ∈ Θ . The assumption of conditional independence may not always be satisfied in reality. For example, the potential recall in the introductory example may very well be correlated with the sales volume of other products of the firm. However, the tradeoff between participation and incentive effect does not hinge on Assumption 1 and the analysis is made tractable. In contrast, the result on justifiability of the optimal contract depends crucially on this assumption. The implications of relaxing Assumption 1 on the optimal contract and its justifiability are discussed in Section 5.1 and Section 5.4 respectively. Awareness structure: Unlike in the standard moral hazard problem, the agent is unaware of some contingencies. The subset the agent is aware of is denoted by Θ A ⊂ Θ . The principal is aware of the entire set Θ . Further, he knows that the agent is unaware and he knows which contingencies the agent is unaware of. The agent is unaware of his unawareness and is unaware of the principal’s superior awareness. When the principal writes the contract he can enlarge the agent’s awareness by mentioning contingencies in the contract, denoted by X ⊆ Θ\Θ A . The agent updates his awareness and  = ΘA ∪ X . considers henceforth all contingencies in the set Θ State spaces: A state of the world in this environment can be thought of as a sequence of 0’s and 1’s of length |Θ| that specifies the realization of each θ ∈ Θ . Let S denote the collection of these sequences. Since the agent is unaware of some contingencies, he does not perceive the actual state space but a less expressive one. A state in the agent’s subjective state  < |Θ| that specifies the realization of each θ ∈ Θ . Let  space can be thought of as a sequence of 0’s and 1’s of length |Θ| S  = {θ1 }. Objectively there are four states of the denote the collection of these sequences. For example, let Θ = {θ1 , θ2 } and Θ world S = {(0, 0), (0, 1), (1, 0), (1, 1)}, but the agent only perceives two  S = {(0), (1)}. In terms of the introductory example, suppose there are two relevant contingencies: the marketing strategy being a success and the product having adverse effects on consumers’ health. If the manager is unaware of the latter, his subjective state space distinguishes between the marketing strategy being a success or a failure but misses the dimension about the adverse health effects. Thus, the set of contingencies the agent is aware of determines the dimension of his subjective state space. Disclosing a contingency in the contract implies adding another dimension to his subjective state space. Outcomes: There is a project with stochastic outcome Y , which is observable and verifiable. Outcome is a function of the contingencies in Θ . Since state s ∈ S specifies the realization of each contingency θ ∈ Θ , we can define outcome realization y directly as a function of the state s ∈ S:

y = f (s),

s ∈ S.

Let Y denote the range of function f . Since the agent does not know the objective state space S, he cannot know Y . Instead he perceives outcome as a function of the contingencies he is aware of. I assume that the agent’s perceived outcome is equivalent to the objective outcome when the contingencies the agent is unaware of are not realized, denoted by θ = 0, implying that the agent considers a subset of possible worlds. Note that θ = 0 can refer to an elementary proposition being true or false.3 A way to think about this assumption is that there are events that the agent has never observed and that have

1 The agent is said to have a positive self-image, whenever there is first-order stochastic dominance of the agent’s perceived distribution over the actual distribution for any action. Santos-Pinto (2008) defines effort and self-image as complements if first-order stochastic dominance is stronger for high effort than for low effort. 2 In my set up the agent implicitly has a positive self-image if he is unaware of outcome decreasing contingencies. If I assume that the true distribution conditional on high effort first-order stochastically dominates the true distribution conditional on low effort and that the agent is unaware of negative outcome shocks only, unawareness implies that effort and positive self-image are substitutes instead of complements. This is because first-order stochastic dominance implies that the probability of low outcomes is more likely under low than high effort. Unawareness implicitly implies that the agent assigns probability zero to some of these outcomes given high and low effort. Thus, the agent’s beliefs deviate stronger for low than high effort. 3 For example, if a decision maker is unaware of the concept of gravity and θ is the elementary statement that there is gravity, then θ = 0 means that the elementary statement is true. Similarly, if a decision maker is unaware of global warming and θ is the elementary statement that there is global warming, then θ = 0 means that the elementary statement is false.

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never crossed his mind. Instead, the agent has some implicit assumptions about the underlying state of the world, but is unaware of these implicit assumptions (Li, 2008). Consequently, he cannot imagine a world in which a proposition implicitly assumed to be true (false) turns out to be false (true). This assumption is prevalent in the literature of unawareness.4 The agent’s outcome function is

y = f (s),

s ∈ S,

 denote the range of function  where  f (s) = f (s, 0, 0, 0, . . .), s ∈  S.5 Let Y f. Assumption 2. |Y | = 2|Θ| . A2 imposes that outcome differs across every state of the world, which implies that the agent knows a subset of possible  ⊂ Y . This implies that the agent is not only unaware of some contingencies outcomes whenever he is not fully aware, Y but also of their consequences. Coming back to the introductory example, the manager is not only unaware of a possible recall but also of its effect on the firm’s performance. If the recall is realized, the manager is surprised and receives the minimal payment. The moral hazard problem to have in mind is one, where unforeseen contingencies lead to unforeseen consequences and the agent may be surprised ex-post. The assumption that f is a one-to-one function is important for tractability of the characterization of the tradeoff between participation and incentives.6 More general outcome functions are discussed in Section 7.1. Probability measures: Let π ( y |e ) denote the probability of y ∈ Y given effort e and assume π ( y |e ) > 0, ∀ y ∈ Y . The distribution over Y is known to the principal. The agent is assumed to have correct beliefs over the distribution of contingencies within his awareness. So whenever the principal expands the agent’s state space by making him aware of a new contingency, the agent understands the conditional probability distribution of this contingency. This belief updating rule implies that the likelihood ratios of events in the original state space remain unchanged and that the relative probability mass assigned to the newly revealed outcomes is correct.7 . Let Π(Θ|  is constant across all y ∈ Y  e ) := /Θ  Under the assumption of independence the probability that θ = 0 for all θ ∈ . Then the agent assigns probability  Pr[θ = 0|e ] denote the probability that y ∈ Y θ∈ /Θ

( y |e ) := π

π ( y |e)

 e) Π(Θ|

,

, y∈Y

 conditional on effort e, which is simply the conditional probability given that none of the unforeseen to outcome y ∈ Y contingencies are realized. The contract: As in the standard principal–agent problem, effort is assumed to be non-observable; thus, the principal offers a contract based on the observable and verifiable outcome Y . The distribution of Y depends on the effort of the agent. It is assumed that E [Y |e H ] > E [Y |e L ] and that E [Y |e H ] − E [Y |e L ] is large enough such that it is always optimal to induce high effort. This allows me to abstract the analysis from the choice of effort. In Section 7.2 the optimal action choice will be discussed. The agent is assumed to have limited liability; thus, the outcome contingent compensation C is non-negative for all y ∈ Y . Due to the Inada conditions imposed on υ (·), the limited liability constraint will not be binding for outcomes within the agent’s awareness. Instead, it implements a lower bound on payments for outcomes the agent is unaware of.  C ) with Θ A ⊆ Θ  ⊆ Θ and C : Y −→ R+ . Definition 1. A contract is a pair (Θ, 0 ∗ ,  Let (Θ C ∗ ) denote the contract that maximizes the principal’s expected payoff. Following Filiz-Ozbay (2012), Definition 2 introduces a notion of incompleteness.  = Θ . Otherwise it is complete.  C ) is incomplete if Θ Definition 2. A contract (Θ, 4 See for example Modica et al. (1998) and Heifetz et al. (2013). Furthermore, the concept of a default dimension is closely related to the assumption of a default action in the literature on unawareness of actions (e.g. Von Thadden and Zhao, 2012a). 5 Assume that any sequence s ∈ S is ordered such that s = {s , s }, with s ∈  S. 6 A2 implies that disclosing contingencies allows the principal to use a more informative outcome distribution, which implies that the effect of disclosure on incentives is always positive. If this assumption is given up, the incentive effect is ambiguous. 7 For a detailed study of belief updating under growing awareness see Karni and Vierø (2012). They derive a belief updating rule that requires that the likelihood ratios of events in the original state space remain unchanged, but their model stays silent about the absolute levels of these probabilities. The belief updating rule in my framework is a special case of the belief updating rule derived by Karni and Vierø (2012). The additional assumption that the relative probability mass assigned to newly revealed states is correct facilitates the exposition of the tradeoff between participation and incentive effect. If this assumption is given up, principal and agent may hold different beliefs about the likelihood of newly revealed contingencies, making the principal want to bet with the agent on the realization of these contingencies. With the assumption that the agent understands the likelihood of contingencies once aware, I abstract from such side bets.

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 is non-empty. The principal can construct the contract such Suppose the proposed contract is incomplete such that Y \Y that he pays zero to the agent when an unforeseen level of outcome is realized, e.g. by finding a functional form of the  or by including a “zero payment otherwise” clause compensation scheme on Y satisfying zero payments for all y ∈ Y \Y in the contract. I will abstract from the question of how the principal can implement zero payments in the unforeseen  are optimal because any positive payment in states, but analyze a reduced form of this model. Zero payments at y ∈ Y \Y these states will leave the agent’s expected utility unaffected, but make the principal strictly worse off. This implies that whenever the contract is incomplete, the agent’s expected utility evaluated at objective beliefs is strictly lower than his reservation utility. It is important to note that zero payments facilitate notation considerably, but that the results hold for any other minimal payment as long as it is low enough.8 Thus, the optimal contract in this environment can be interpreted as a contract that promises a fixed payment and that rewards the agent with bonuses for certain outcomes. Whenever a contingency is realized that is not anticipated by the agent, the bonus is not paid. Expected payoffs: The principal’s outside option in the case of rejection is assumed to be zero. His expected payoff is given by

 EU P =

0

y ∈Y

π ( y |e)[ y − C ( y )] if the agent accepts, if the agent rejects.

The agent assesses his expected utility with respect to his restricted state space. The outside option of rejecting the contract is U¯ :



EU A =

 ( y |e ) y ∈Y

υ (C ( y )) − e if the agent accepts,

π

U

if the agent rejects.

4. The optimal contract with observable effort In order to have a benchmark it is useful to first characterize the contract when effort is observable. If effort is observable and verifiable the contract can be made directly contingent on the action of the agent. The principal solves the problem:

max

 C (·) Θ,

 





π y |e H y − C ( y )

y ∈Y

subject to

      y |e H υ C ( y ) − e H  U , π  y ∈Y

C ( y )  0,

∀y ∈ Y.

When the contract is complete, it is optimal to give the agent full insurance. This can be seen from the first order conditions

1

υ  (C ( y ))

= λ,

∀y ∈ Y.

The agent receives C F B = υ −1 (U¯ + e H ) independent of the realization of Y . The first best is achieved. Now suppose the  are principal leaves the agent unaware of some contingencies. The first-order conditions for C ( y ), y ∈ Y

1

υ  (C ( y ))

=λ

1

 eH ) Π(Θ|

.

 is constant. The optimal compensation scheme is simply The first-order conditions imply that the transfer across y ∈ Y  and  . The expected payment to the agent is Π(Θ|   e H )C F B . It is minimized C ∗ ( y) = C F B , ∀ y ∈ Y C ∗ ( y ) = 0, ∀ y ∈ Y \Y  e H ), is minimized, which is achieved when the agent’s awareness level is lowest. when the probability of paying, Π(Θ| Consequently, if effort is observable, it is optimal to reveal nothing to the agent.

∗ = Θ A . Proposition 4.1. Under A1, A2 and observable effort, Θ The reason for result Proposition 4.1 is that disclosing contingencies to the agent makes the participation constraint more costly to satisfy. Since effort is observable there is no incentive effect and only the participation effect matters.

8 Low enough means that the minimal payment constraint is not binding for outcomes within the agent’s awareness. Otherwise the tradeoff for the respective outcomes changes.

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5. The optimal contract with unobservable effort If effort is unobservable, the principal maximizes his expected profit subject to the participation constraint, the incentive constraint and the limited liability constraints. The participation constraint assures that the agent accepts the contract. The incentive constraint leads the agent to exert high effort e H . The principal solves:

max

 C (·) Θ,

 





π y |e H y − C ( y )

(1)

y ∈Y

subject to

      y |e H υ C ( y ) − e H  U , π

 y ∈Y



H

e ∈ arg max e

C ( y )  0,

(2)

  ( y |e )υ C ( y ) − e , π 

(3)

 y ∈Y

∀y ∈ Y,

(4)

where (2) is the participation constraint, (3) is the incentive constraint and (4) is the limited liability constraint. Assume that a solution to the maximization problem exists.9 The analysis of the optimal contract can be divided into two steps. In . In step two he chooses the step one the principal chooses the optimal compensation scheme  C given announcement Θ . optimal level of awareness Θ

 5.1. Step 1: Optimal compensation scheme given awareness Θ  is characterized by the necessary condition The optimal compensation scheme given awareness Θ 1

υ  (C ( y ))

=



( y |e L ) π , λ + γ 1−  eH ) ( y |e H ) π Π(Θ|

1

, ∀y ∈ Y

(5) ( y |e L ) π

.10 Note that the optimal compensation scheme varies with the likelihood ratio as well as C ( y ) = 0, ∀ y ∈ Y \Y of ( y |e H ) π   the restricted information structure Θ instead of Θ . Since the agent is unaware of the contingencies in Θ\Θ , these signals cannot be used to induce e H . As in the standard moral hazard problem both participation and incentive constraint hold C denote the solution to this system of equations. with equality. Let  Lemma 5.1. Assume A1 and A2. Under  C , both λ > 0 and γ > 0.

2

Proof. See Appendix A.2.2.

Remark. If the assumption of conditional independence is relaxed, the optimal compensation scheme additionally varies ( y |e H ) π

( y |e H ) π

.11 Under Assumption 1, , but if conditional dependence is is constant across all y ∈ Y with the ratio π ( y |e H ) , y ∈ Y π ( y |e H ) not ruled out, this ratio generally varies across outcomes. In such a case, it is optimal to promise relative large payments for outcomes that are positively correlated with contingencies the agent is left unaware of, because these payments are less likely to be realized.  coincides with the optimal compensation scheme of the standard The optimal compensation scheme  C (·) across y ∈ Y . To see this, suppose that  S is the principal–agent model with symmetric awareness and restricted information structure Θ objective state space and that both, the principal and the agent, are symmetrically aware of  S. Then the principal solves min C (·)

    y |e H C ( y ) π

(6)

 y ∈Y

subject to (2) and (3). Let C C denote the solution to this problem. C C is the optimal complete contract under symmetric Θ

Θ

. Under asymmetric awareness, the expected payment to the agent is awareness and information structure Θ 9 10 11

For details see Grossman and Hart (1983). . Inada conditions assure that (4) is not binding for C ( y ), y ∈ Y If Assumption 1 is relaxed, the first-order condition is

1

υ  (C ( y ))



=

( y |e L ) ( y |e H ) π π λ+γ 1− ( y |e H ) π ( y |e H ) π

 ,

. ∀y ∈ Y

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S. Auster / Games and Economic Behavior 82 (2013) 503–521

 





 eH π y |e H C ( y ) = Π Θ|

   H    · 0,  y |e C ( y ) − 1 − Π(Θ) π  y ∈Y

y ∈Y

, the  e H ). Since Π(Θ|  e H ) is nothing but a constant for a given Θ which is equivalent to (6) except for the scaling factor Π(Θ| C   two optimization problems are equivalent and C ( y ) = C  ( y ) for all y ∈ Y . The expected profit of the optimal incomplete Θ

contract is simply the expected payment of C C weighted by the probability that none of the unforeseen contingencies are Θ realized:











 e H E C C (Y )|e H . E  C (Y )|e H = Π Θ| Θ ∗ 5.2. Step 2: Optimal awareness Θ The optimal level of disclosure is characterized by identifying the basic tradeoff between participation and incentives of disclosing contingencies to the agent. To separate the effect on incentives from the effect on participation, it is useful to compare the expected payment of complete contracts under different information structures.

. Then Lemma 5.2. Let Z be a non-empty subset of Θ\Θ

C

C

H H − E C Θ∪  0, C ΘZ := E C Θ  (Y )|e  Z (Y )|e with strict inequality if and only if ∃θ ∈ Z such that Pr[θ = 1|e H ] = Pr[θ = 1|e L ].

2

Proof. See Appendix A.2.3.

This result is in line with Holmström’s Sufficient Statistic Theorem (1979), which states that a signal θ is valuable if and only if it is informative.12 Valuable means that both, principal and agent, can be made better off by including θ because agency costs are reduced. Under independence, θ is informative if and only if Pr[θ = 1|e H ] = Pr[θ = 1|e L ]. 5.3. The basic tradeoff To understand the effect of disclosing a subset of Θ\Θ A on participation and incentives, compare the expected payoff of  = Θ A ) and revealing set X (Θ  = Θ A ∪ X ): the principal when revealing nothing (Θ

 C

  C

 Π Θ A |e H E C Θ (Y )|e H ≷ Π Θ A ∪ X |e H E C Θ (Y )|e H , A A∪X

(7)

or simply



θ∈X











C C Pr θ = 0|e H E C Θ (Y )|e H ≷ E C Θ (Y )|e H . A A∪X

C C X Using C Θ = E [C Θ (Y )|e H ] − E [C Θ (Y )|e H ] we can restate (7) in terms of gains and losses of revealing set X : A A A∪X



X

C Θ A ≷ 1 −







Pr θ = 0|e

θ∈X

H







C E CΘ (Y )|e H . A

(8)

C ΘX A captures the incentive effect of disclosing set X . The announcement of a set of informative contingencies allows the  X . (1 − θ∈ X Pr[θ = principal to use these contingencies as a signal about the agent’s effort. This information gain is C Θ A  C 0|e H ]) E [C Θ (Y )|e H ] captures the participation effect of disclosing X . With probability 1 − θ∈ X Pr[θ = 0|e H ] one of the A contingencies in X is realized. When announcing Θ A ∪ X , the principal has to pay the agent a positive wage, while when announcing Θ A , he pays zero. Disclosing a contingency to the agent consequently tightens the participation constraint. When effort is observable, there is no incentive effect, which is why it is optimal to keep the agent unaware. When effort  such that the net gain of revelation is maximized. Since Θ is finite, he compares is unobservable, the principal chooses Θ  such that the a finite number of announcement strategies and their respective expected payoffs. Whenever there exists a Θ incentive effect outweighs the participation effect, the principal enlarges the agent’s awareness. To illustrate the basic tradeoff consider the announcement of a single contingency θ ∈ / ΘA : θ C Θ A

vs.







C Pr θ = 1|e H E C Θ (Y )|e H . A

Fig. 1 shows the incentive gain and the participation loss as a function of Pr[θ = 1|e H ].13 12

Holmström (1979) shows this for continuous outcome and continuous effort. 1−σ

θ C This figure shows C Θ and Pr[θ = 1|e H ] E [C Θ (Y )|e H ] for the following specification: υ (C ) = C1−σ , σ = 0.5, e H = 1, e L = 0, U¯ = 5. There are two A A H contingencies. Contingency θ ∈ Θ A with Pr[θ = 1|e ] = 0.65 and Pr[θ = 1|e L ] = 0.5 and contingency θ  ∈ / Θ A with Pr[θ  = 1|e L ] = 0.5. 13

S. Auster / Games and Economic Behavior 82 (2013) 503–521

511

Fig. 1. Tradeoff.

θ is monotonically increasing in θ . Lemma 5.3. Assume A1 and A2. Let θ := | Pr[θ = 1|e H ] − Pr[θ = 1|e L ]|. C Θ A

Proof. See Appendix A.2.4

2

θ θ is a measure of informativeness of signal θ . Lemma 5.3 states that the incentive gain C Θ is increasing in the A L informativeness of θ . Consequently, the incentive gain is decreasing on the interval [0, Pr[θ = 1|e ]) and increasing on the interval (Pr[θ = 1, e L ], 1]. When Pr[θ = 1|e L ] = Pr[θ = 1|e H ], the signal is uninformative and the incentive gain is zero. C (Y )|e H ] is linearly increasing in Pr[θ = 1|e H ]. Fig. 1 shows that it is optimal to The participation loss Pr[θ = 1|e H ] E [C Θ A reveal any informative contingency θ if the probability that θ = 1 conditional on high effort is small enough. This result is

summarized in Observation 5.4. Observation 5.4. Assume A1 and A2. For every θ ∈ Θ\Θ A there exists a threshold < π θ , the incentive effect outweighs the participation effect.

π θ ∈ (0, Pr[θ = 1|e L ]) such that if Pr[θ = 1|e H ]

Example 5.1. The tradeoff associated to the announcement of a single contingency generally depends on the announcement of other contingencies. Going back to the introductory example, suppose the manager is not only unaware of a possible lawsuit but also of a possible merger. Whether it is profitable to reveal the merger not only depends on its likelihood conditional on the manager’s effort, but also on whether the firm reveals the lawsuit or not. If the lawsuit is revealed and its realization is highly informative about the manager’s effort, the incentive effect of revealing the possibility of a merger is negligible. Thus, the tradeoff between incentives and participation associated to the announcement of a single contingency generally depends on the announcement of other contingencies. To see how the single contingency tradeoff changes as the underlying awareness extends, consider the example of N symmetric contingencies, Θ\Θ A = {θ1 , . . . , θ N }, where symmetric n := Θ A ∪ θ1 ∪ · · · ∪ θn .14 means that every contingency has the same distribution. Let α := Pr[θi = 0|e H ], i = 1, . . . , N and Θ The net gain of revealing the nth contingency is:





C H N G (θn ) := α N −n C n − (1 − α ) E C Θ n−1 (Y )|e Θn−1

θ



.

The value of N G (θn ) is determined by two factors. First, there is the difference between incentive gain and participation loss as illustrated above. This difference is weighted by the probability that the gains and losses are realized, α N −n . Generally, N G (θn ) can decrease or increase in n. A sufficient condition for N G (θn ) to be decreasing in n is:



C H E CΘ n (Y )|e

2

C < E CΘ 

n −1

C (Y )|e H E C Θ 

n +1

(Y )|e H ,

for all n ∈ {1, . . . , N − 1}.

(9)

Condition (9) implies that the expected payment of the complete contract C C as a function of n is sufficiently convex, i.e. Θn the inclusion of signal θn in the information structure reduces agency costs strongly when n is small but only marginally when n is large.15 If this condition holds, the net gain of revealing a contingency is decreasing in the number of other

14

0 = Θ A . For notational convenience let Θ

15

Condition (9) is satisfied, for example when

υ (C ) =

C 1−σ 1−σ

,

σ = 0.5, e H = 1, e L = 0, U¯ = 9, N = 5, α = 0.6 and Pr[θ = 0|e L ] = Pr[θ = 1|e H ] = 0.5.

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∗ = {Θ A , θ1 , . . . , θn∗ } such that: contingencies that are revealed and the optimal level of disclosure is Θ N G (θn∗ )  0 and

N G (θn∗ +1 ) < 0.

5.4. Justifiability of the contract If the contract is incomplete, the agent’s perception of the world differs from the perception of the principal. An impor C) tant question is whether the proposed contract can elicit suspicion on the side of the agent. Upon receiving contract (Θ,  and compensation scheme C are optimal for the principal. Filiz-Ozbay the agent may ask herself whether announcement Θ (2012) introduces an equilibrium refinement which requires that the equilibrium contract maximizes the principal’s expected payoff from the viewpoint of the agent. The agent can only contemplate contracts within his awareness. The set of ˜ C˜ ) such that Θ A ⊆ Θ˜ ⊆ Θ . Let E ˜ and Y˜ denote the contracts the agent is aware of is given by the set of all contracts (Θ, Θ

 C ) is called justifiable if is optimal expectation operator and the set of outcomes associated to awareness Θ˜ . A contract (Θ, for the principal from the viewpoint of the agent, both in terms of compensation and in terms of the level of disclosure. , let C ˜ : Y → R+ denote the solution to Definition 3. For any Θ˜ such that Θ A ⊆ Θ˜ ⊆ Θ 0 Θ max C

s.t.



E Θ˜ Y − C (Y )|e E Θ˜







υ C (Y ) |e − e  U , 

e ∈ arg max E Θ˜ e˜







υ C (Y ) |˜e − e˜ ,

 C ) is called justifiable if and C ( y ) = 0, ∀ y ∈ Y \Y˜ . A contract (Θ,



 ∈ arg max E Θ Y − C ˜ (Y )|e and C ( y ) = C Θ ( y ), ∀ y ∈ Y . Θ Θ Θ˜

{C Θ˜ }Θ A ⊆Θ⊆ ˜ Θ  is the set of justifiable compensation schemes for each level of disclosure the agent can contemplate.

 C ) is called justifiable if it satisfies two requirements. First, compensation scheme C has to be justifiable for A contract (Θ,  and second, from the viewpoint of the agent, compensation scheme C is the best of all justifiable compenawareness Θ sation schemes in the set {C Θ˜ }Θ ⊆Θ⊆ ˜ Θ  . Note that a justifiable contract generally exists, because if none of the incomplete A

C contracts is justifiable, the principal can always resort to the full awareness contract (Θ, C Θ ). Since full disclosure implies C that the principal and the agent share the same beliefs, (Θ, C Θ ) is necessarily justifiable. Thus, provided that there is a solution to the principal’s optimization problem under symmetric awareness Θ , a justifiable contract exists. Since the optimal ∗ ,  contract (Θ C ∗ ) is not necessarily complete, the question arises under which conditions the optimal contract is justifiable for the agent.

H ∗ ,  Proposition 5.5. Assume A1 and A2. (Θ C ∗ ) is justifiable according to Definition 3 if and only if E Θ ∗ [ Y − C Θ ∗ (Y )|e ]  0 and H L − 1 L ¯ , where C¯ = υ (U + e ). EΘ ∗ [ Y − C Θ ∗ (Y )|e ]  E Θ ∗ [ Y |e ] − C

Proof. See Appendix A.2.5.

2

A necessary and sufficient condition for the optimal contract to be justifiable is that the principal’s expected utility is non-negative and that e H is the optimal action choice from the viewpoint of the agent. Note that in the characterization of the optimal contract, actual outcome levels in Y play no role because only the likelihood ratio associated with each outcome level is relevant for providing of incentives. If we require the contract to be justifiable, this is no longer necessarily the case. Whenever the optimal contract is incomplete, the agent perceives only a subset of possible outcomes. It is possible that the H H optimal contract leaves the agent unaware of high outcomes such that E Θ ∗ [ Y − C Θ ∗ (Y )|e ] < 0 whereas E [ Y − C Θ ∗ (Y )|e ]  0. Similarly, it is possible that the optimal contract leaves the agent unaware of outcomes that are strongly correlated H L ¯ whereas E [Y − C Θ∗ (Y )|e H ]  E [Y |e L ] − C¯ .16 If any of the two with effort such that E Θ ∗ [ Y − C Θ ∗ (Y )|e ] < E Θ ∗ [ Y |e ] − C conditions in Proposition 5.5 is violated, the refinement introduces another dimension in the tradeoff. The principal is no longer only concerned with the distributional properties of the contingencies in Θ\Θ A , but also with the outcomes an announcement reveals. 16 An example for the first violation is a potential innovation that the manager is unaware of without which the firm is not profitable. It seems more realistic that firms stay silent about bad news rather than good news, which may have reasons not captured in this model (e.g. restriction to monotone compensation schemes). The second violation requires that the agent is unaware of an event that is sufficiently correlated with his effort and that is sufficiently likely, e.g. the success of an advertisement campaign. It seems more realistic that decision makers are unaware of low probability events rather than events that occur frequently, so the economic relevance of both violations may be restricted.

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513

Given that the expected profit evaluated at the agent’s beliefs is non-negative and higher at e H than at e L , the optimal ∗ ,  C ∗ ) is justifiable. Justifiability of  C ∗ is straight forward. If the optimal contract is complete, principal and contract (Θ C ∗ maximizes the principal’s expected payoff from the agent’s perspective. If the agent share the same beliefs. Hence,  optimal contract is incomplete, we know that the transfer rule for outcomes within the agent’s awareness coincides with . Since the agent thinks that the the optimal compensation scheme of the complete contract given information structure Θ C ∗ solves the principal’s optimization problem given the agent’s beliefs. contract is complete,  ∗ , remember that the agent can only consider the announcement of contingencies within his To see justifiability of Θ ∗ \ Z , where ∗ . The agent evaluates the principal’s expected payoff for every announcement Θ awareness. Suppose Θ A ⊂ Θ C ∗ ∗   Z ∈ Θ \Θ A . The agent knows that given announcement Θ \ Z the optimal compensation scheme is C ( y ) = C ∗ ( y ) for all Θ \Z

∗ \ Z and zero otherwise. Thus, after reading the contract, the agent outcomes within the agent’s hypothetical awareness Θ understands the principal’s optimal contract for any level of awareness lower or equal than his actual awareness, but he ∗ to be justifiable, the following does not understand that there may remain contingencies that he is unaware of. For Θ condition has to be satisfied







C H  EΘ  CΘ ∗ (Y )|e

∗ \ Z θ ∈Θ









C H Pr θ = 0|e H E Θ  CΘ ∗ \ Z (Y )|e ,

(10)

∗ can ∗ \Θ A . This coincides with the optimality condition of the principal. Consequently, (10) is fulfilled and Θ for any Z ⊆ Θ be rationalized by the agent. Note that the assumption of conditional independence is crucial for justifiability of the optimal contract. If this assump( y ) π , which cannot be rationalized by the tion is not satisfied the optimal compensation scheme varies with the ratio π ( y ) , y ∈ Y agent. If we assume that the agent rejects any non-justifiable contract, this implies that the principal proposes the contract ∗ }, in order to ensure acceptance by the agent. that is optimal under conditional independence, { C ∗, Θ Chen and Zhao (2009) propose two additional equilibrium concepts: the trap-filtered equilibrium and the trap-filtered equilibrium with cognition. In the trap-filtered equilibrium the agent assigns some probability ρ to the event that a nonjustifiable contract is a trap and probability 1 − ρ to the event that the contract is a consequence of the principal’s mistake. If the contract is a trap the agent is better of rejecting it, while if the contract is proposed by mistake, the H agent prefers to accept it. Applying this equilibrium concept to my framework implies that, given E Θ ∗ [ Y − C Θ ∗ (Y )|e ] < 0 H L ¯ or E Θ ∗ [ Y − C Θ ∗ (Y )|e ] < E Θ ∗ [ Y |e ] − C , the principal finds it optimal to propose a non-justifiable contract if and only if the agent’s belief that the contract is a trap is sufficiently low. In the trap-filtered equilibrium with cognition the agent can exert cognitive effort to learn about the probability that the contract is a trap. Cognitive effort is costly, so the agent chooses a level such that the marginal cost of cognitive effort equals its marginal information gain. When designing the contract the principal takes the agent’s cognitive effort choice into account. Depending on the agent’s prior and the cognitive cost function, the principal offers either a justifiable contract that is accepted with probability one or he offers a non-justifiable contract that is only accepted when the agent does not learn that the contract is a trap. 6. Competing principals In the basic setup I analyze the optimization problem of a monopolistic principal. This section addresses the question of how these results change when principals compete against each other. Suppose there are N principals that are aware i , C i ), i = 1, . . . , N. The agent updates his awareness after hearing all of Θ . They make simultaneous offers, denoted by (Θ 1 ∪ · · · ∪ Θ N . If the agent is indifferent the offers and accepts at most one. He considers henceforth every contingency in Θ between two or more contracts he accepts each contract with equal probability. I focus on symmetric equilibria in pure i , C i ) = (Θ,  C ), i = 1, . . . , N. strategies, (Θ In the absence of asymmetric awareness, principals engage in a Bertrand competition over transfer rule C . In equilibrium they make zero profits and the surplus goes to the agent. Assume that under full awareness the agent’s expected payoff is maximized when high effort is implemented and let C ∗ denote the equilibrium compensation scheme. Assumption 3. Let E [υ (C ∗ ( y ))] − e H  U and E [υ (C ∗ ( y ))] − e H  υ ( E [Y |e L ]) − e L , where C ∗ maximizes the aware agent’s payoff subject to his incentive constraint and the zero profit constraint. Proposition 6.1. Assume A1, A2 and A3. There is a symmetric Nash equilibrium in which the agent is fully aware and each principal offers the complete zero profit contract (Θ, C ∗ ). Proof. See Appendix A.2.6.

2

Proposition 6.1 states that the full awareness equilibrium exists.17 To see this, note that whenever the announcements of the other principals promote full awareness the own announcement is not payoff relevant. Further, any change in the

17

Filiz-Ozbay (2012) derives a similar result in her framework.

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compensation scheme yields either negative or zero expected profits for the standard Bertrand competition argument. Thus, the full awareness equilibrium exists. In general there can be other equilibria in which the agent is not fully aware. To see this, consider the following example. Example 6.1. Suppose there are two contingencies, Θ = {θ1 , θ2 }, of which the agent is aware of one, Θ A = {θ1 }. Assume that θ1 is not informative about the agent’s action, i.e. Pr[θ1 = 0|e L ] = Pr[θ1 = 0|e H ]. There are two possible equilibria, one in which all principals reveal θ2 and one in which the principals leave the agent unaware. In the latter there is no incentive compatible compensation scheme, so the equilibrium action is e L . The principals propose a fixed payment C¯ for all outcomes within the agent’s awareness and pay zero otherwise, Whether the unawareness equilibrium exists or not depends on the distribution of θ2 . If Pr[θ2 = 0|e H ]  Pr[θ2 = 0|e L ], θ2 is always revealed and the full awareness equilibrium is unique. To see this, suppose (θ1 , C¯ ) is the equilibrium contract and consider a deviation by principal i, making the agent aware of contingency θ2 and splitting the second-best surplus between them. The agent updates his awareness and chooses between the equilibrium contract and the deviating contract. If he chooses the equilibrium contract, the agent exerts low effort and obtains payoff Pr[θ2 = 0|e L ]υ (C¯ ) − e L . Since C¯ is feasible but not optimal under symmetric awareness, the deviating principal can find a compensation scheme that is strictly preferred by the agent and yields a positive profit. Thus, the full awareness equilibrium is unique. On the other hand, if Pr[θ2 = 0|e H ] > Pr[θ2 = 0|e L ], the equilibrium in which all principals offer (θ1 , C¯ ) and the agent stays unaware may exist. C¯ is determined by the zero profit condition of the principals:









E Y |e L − Pr θ2 = 0|e L C¯ = 0. Note that the zero profit condition implies that C¯ is large if the probability that the principals have to pay, Pr[θ2 = 0|e L ], is small. Now suppose principal i deviates by offering a complete contract (Θ, C˜ ). The agent chooses between the deviating contract and the equilibrium contract. In contrast to the previous case the agent may find it optimal to adjust his action choice when accepting the incomplete equilibrium contract, because the probability of receiving C¯ is increasing in effort. If the agent accepts the equilibrium contract (θ1 , C¯ ) and exerts high effort, principals offering the equilibrium contract make a loss and the agent obtains payoff



Pr θ2 = 0|e H



υ (C¯ ) − e H .

If Pr[θ2 = 0|e L ] is sufficiently small such that C¯ is sufficiently large, this payoff exceeds the surplus of the incentive compatible complete contract (Θ, C ∗ ).18 In this case there is no contract that attracts the agent and yields a positive expected payoff, making the deviation unprofitable. Thus, the unawareness equilibrium exist.19 The intuition why competition may not necessarily lead to full revelation is that if the agent’s perception of the world is sufficiently distorted, principals can make “generous” offers for outcomes the agent is aware of. Making the agent aware allows the agent to adjust his action choice, exploiting the equilibrium offer in his favor at the cost of the principals. Since this argument does not depend on the numbers of principals, even under intense competition there may exist equilibria in which the agent is unaware and the constrained-efficient action is not implemented.20 The possibility of unawareness despite competition in this setting is closely related to the markets with shrouded attributes as described in Gabaix and Laibson (2006). The authors analyze a competitive market setting in which firms offer a product that has hidden add-on prices of which some consumers are unaware.21 Consumers can avoid paying for the add-on by exerting substitution effort before purchasing the basic good, assumed to be inefficient. Firms choose a price for the basic good and a price for the add-on. There exists an efficient equilibrium in which firms set the price for the add-on low enough such that all consumers, aware or unaware, find it optimal to purchase the add-on instead of exerting substitution effort. However, if the share of unaware types is large enough, there exists another inefficient equilibrium in which the add-on is shrouded and unaware consumers are exploited by the firms. In this equilibrium firms offer a low price for the basic good and charge a high price for the add-on. A consumer that is aware of the add-on optimally exerts substitution effort prior to purchasing the good while an unaware consumer is forced to pay the high price for the add-on.

18

This is the case if



Pr θ2 = 0|e H



υ



E [Y |e L ]

Pr[θ2 = 0|e L ]



 

 E υ C ∗ |e H .

19 Going back to the introductory example, suppose there is more than one firm and suppose all firms are aware of their product’s effect on consumer health. In the unawareness equilibrium, firms do not reveal possible adverse health effects to the manager. The manager, not taking this possibility into account, exerts too little effort and may be surprised ex-post. 20 Note that there may also be equilibria in which the agent is unaware and exerts high effort. In such an equilibrium, upon becoming aware, the agent exploits the principals’ equilibrium offer in his favor by exerting low effort. 21 Gabaix and Laibson (2006) call their consumers myopic and non-myopic instead of aware and unaware, but the spirit is the same.

S. Auster / Games and Economic Behavior 82 (2013) 503–521

515

If the unaware consumer is made aware, he can profit from the low price for the basic good and if this price is low enough, the efficient contract cannot attract any consumer. Just as in the competition environment in my setting, there may exist an inefficient equilibrium with unawareness in which the competitive effect is overturned by a “curse of debiasing” (Gabaix and Laibson, 2006): disclosing the features of the inefficient equilibrium contract makes this contract more attractive. Such a curse arises when the contract that takes advantage of the unaware type may be taken advantage of by the aware type. 7. Discussion 7.1. The output function In the basic model, output is discrete and differs across every state of the world. If y is not one-to-one and contingencies are not observable, both participation loss and incentive gain of revelation are affected. The participation loss is generally diminished, because the revelation of an unforeseen contingency does not necessarily imply the revelation of an unforeseen outcome. The incentive gain is no longer unambiguous, because the agent’s perceived distribution of outcome can be more informative about the agent’s effort than the actual distribution of outcome. To see this, suppose there are only two possible realizations of outcome. The project can be either a success or a failure,  ⊂ Θ the agent Y = {s, f } with s > f . Whether the project is a success or a failure depends on the realization of Θ . If Θ (.|e ). Now consider the tradeoff the principal faces when is aware of outcomes {s, f } but believes probability distribution π / Θ A . Since the revelation of θ does not reveal any new outcomes, the participation loss of the announcement disclosing θ ∈ is zero. The effect on incentives depends on how the revealed contingency affects the perceived distribution of the agent. To see that the incentive effect can be negative, suppose Θ = {θ1 , θ2 }, Θ A = {θ1 } and assume θ2 is not informative, i.e. Pr[θ2 = θ = 0, because the principal has the choice 1|e H ] = Pr[θ2 = 1|e L ]. In the basic model Pr[θ = 1|e H ] = Pr[θ = 1|e L ] implies C Θ A to ignore the realization of θ2 . Under Y = {s, f } this is no longer the case. Suppose that y = s whenever θ1 = θ2 = 0 and ∗ = Θ A is smaller than the y = f otherwise. Solving for the optimal compensation scheme, the expected payment under Θ ∗ 22  expected payment under Θ = Θ . There is an incentive loss of revealing θ2 because the perceived distribution of the unaware agent is more informative about the action choice than the true distribution.23 7.2. Optimal action choice Throughout the analysis I assumed that E [Y |e H ] − E [Y |e L ] is large enough such that the principal always finds it optimal to induce e H . Unawareness makes incentives more costly, hence it is generally possible that for different levels of awareness different levels of effort are optimal. Since effort can only be high or low, the analysis of the optimal contract under low effort is straightforward.

 Proposition 7.1. Assume A1 and A2. If e L is the action choice, the optimal contract is (Θ A , C L ) with C L ( y ) = υ −1 (U¯ − e L ), ∀ y ∈ Y . and C L ( y ) = 0, ∀ y ∈ Y \Y 2

Proof. See Appendix A.2.7.

The optimal contract inducing low effort leaves the agent unaware because there is no incentive effect. The principal induces low effort in equilibrium if







E Y |e L −



Pr θ = 0|e L











∗ θ ∈Θ\Θ

θ ∈Θ\Θ A

22





υ −1 U¯ − e L > E Y |e H −





C H Pr θ = 0|e L E C Θ ∗ (Y )|e .

If the principal leaves the agent unaware, the optimal compensation scheme is



C u (s) = υ −1 U¯ +

Pr[θ1 = 1|e L ]e H − Pr[θ1 = 1|e H ]e L Pr[θ1 =

1|e L ] − Pr[θ

1

=

1|e H ]

 ,



C u ( f ) = υ −1 U¯ −

Pr[θ1 = 0|e L ]e H − Pr[θ1 = 0|e H ]e L Pr[θ1 =

1|e L ] − Pr[θ

1

=

1|e H ]

 .

If the principal reveals contingency θ2 , the optimal compensation scheme is



(1 − Pr[θ1 = 0|e L ] Pr[θ2 = 0|e L ])e H − (1 − Pr[θ1 = 0|e H ] Pr[θ2 = 0|e H ])e L Pr[θ1 = 0|e H ] Pr[θ2 = 0|e H ] − Pr[θ1 = 0|e L ] Pr[θ2 = 0|e L ]  Pr[θ1 = 0|e L ]e H − Pr[θ1 = 0|e H ]e L . C a ( f ) = υ −1 U¯ − L H Pr[θ1 = 1|e ] − Pr[θ1 = 1|e ] C a (s) = υ −1 U¯ +

So C u ( f ) = C a ( f ) and 23

[θ2 =1|e ] υ (C u (s)) − υ (C a (s)) = − Pr Pr[θ =0|e H ] H

2

e H −e L Pr[θ1 =0|e H ]−Pr[θ1 =0|e L ]



< 0, which implies that E [C u (Y )|e H ] < E [C a (Y )|e H ].

When the agent is aware of all outcomes as in the example, the problem is closely related to a framework without unawareness but with heterogeneous priors. A further discussion on heterogeneous priors and moral hazard can be found in Santos-Pinto (2008) and De la Rosa (2011).

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Whether this is the case or not depends on the distributional properties of the random variable Y , but it is easy to find examples where e H is the optimal action choice under full awareness and e L is the optimal action choice under asymmetric awareness. 7.3. Heterogeneous agents In the basic model it is assumed that the principal knows the agent’s awareness Θ A . Suppose now that awareness is private information and let β denote the principal’s prior on the event that the agent is fully aware. Consider first the case of a monopolistic principal. To make things interesting assume that the optimal contract for the unaware type is incomplete. The principal cannot screen the agent’s type, because this would require specifying non-zero payments for outcomes the unaware type is unaware of, which consequently makes him aware.24 Hence, the principal either offers the optimal incomplete contract, which is rejected by the aware type, or the optimal complete contract, which is accepted by all types. The first option is optimal if the probability that the agent is unaware is large enough. This is in line with Von Thadden and Zhao (2012a), who show that in populations with a large extent of unawareness, contracts are incomplete.25 Next, suppose that there are N principals competing with each other. The results derived in Section 6 extend to this environment. There exists a full awareness equilibrium in which principals reveal all contingencies and make zero profits in expectation. Furthermore, there may exist equilibria in which the unaware type stays unaware, even under intense competition. In such equilibria the aware type is cross-subsidized by the unaware type. To see this consider Example 6.1 but assume that the agent is aware of contingency θ2 with probability β . Consider an equilibrium in which principals offer an incomplete contract which promises a fixed transfer C¯ for all outcomes the unaware type is aware of and zero otherwise. Suppose that C¯ is large enough such that the contract is accepted by both types and such that the aware type finds it optimal to exert high effort. The unaware type does not take the zero payments into account and exerts low effort. Principals do not know whether the agent is aware or unaware and their payoff depends the agent’s action choice. Ex-ante, the equilibrium contract yields payoff:







   β E Y |e H − Pr θ2 = 0|e H C¯ + (1 − β) E Y |e L − Pr θ2 = 0|e L C¯ . The zero-profit condition implies

C¯ =

β E [Y |e H ] + (1 − β) E [Y |e L ] . β Pr[θ2 = 0|e H ] + (1 − β) Pr[θ2 = 0|e L ]

If the share of unaware types is large enough and Pr[θ2 = 0|e L ] is small enough such that C¯ is large enough, there is no complete contract that makes the aware type better of and yields a non-negative expected payoff.26 Thus, no principal has incentives to deviate and the unawareness equilibrium exists. Note that condition (11) implies that principals make losses on the aware type while they make positive profits on the unaware type. In a market with a population of agents this implies that aware types are cross-subsidized by unaware types, just as in Gabaix and Laibson (2006). Such cross-subsidization between types can occur in equilibrium only if there are sufficiently many unaware agents. Thus, the inefficient equilibrium exists only if the extent of unawareness in the population is large enough. 7.4. The agent as the residual claimant The basic model assumes that the principal is the residual claimant. In the presence of asymmetric awareness, ownership of the project matters, because the agent’s valuation of the project depends  on his level of awareness. If the agent is the ( y |e H )υ ( y ). The principal solves: residual claimant, unawareness generally affects his perceived outside option π y ∈Y

max

 C (·) Θ,

 





π y |e H P − C ( y )

y ∈Y

subject to

24 This assumes that the principal wants to induce high effort for both types. However, it is possible that the principal offers a (menu of) incomplete contract(s) and the two types choose different actions. This can only be optimal if β is small enough. 25 In Von Thadden and Zhao (2012a) the principal can screen the types, which comes at a cost of not incentivizing the aware type optimally, because otherwise he pretends to be unaware. If the share of aware types is small, this cost is small and it is optimal to offer the screening menu that leaves the unaware agent unaware. 26 This is the case if



Pr θ2 = 0|e H



υ



β E [Y |e H ] + (1 − β) E [Y |e L ] β Pr[θ2 = 0|e H ] + (1 − β) Pr[θ2 = 0|e L ]



 

 E υ C ∗ |e H .

(11)

S. Auster / Games and Economic Behavior 82 (2013) 503–521

517

       H  y |e H υ y + C ( y ) − P   y |e υ ( y ), π π  y ∈Y



e H ∈ arg max e

C ( y )  0,

 y ∈Y







( y |e )υ y + C ( y ) − P − e , π

 y ∈Y

∀y ∈ Y,

where P is the premium paid by the agent and C ( y ) is the outcome contingent transfer.27 As in the basic model, revealing a contingency θ ∈ / Θ A involves a tradeoff between participation and incentives. In addition, enlarging the agent’s awareness affects his perceived outside option. This effect is favorable to the principal if

       y |e H υ ( y ), π˜ y |e H υ ( y ) < π y ∈Y˜

 y ∈Y

 ∪ θ , θ ∈ Θ\Θ . This is the case if E [Y |θ = 1] < E [Y |θ = 0], i.e. if θ = 1 is a negative outcome shock. Consewhere Θ˜ = Θ quently, if the agent is the residual claimant, not only do the distributional properties of θ ∈ / Θ A matter but also the outcome its announcement reveals. Roughly speaking, when the agent is the residual claimant the principal includes contingencies in the contract that are very unlikely, highly informative and that reveal “bad” outcomes. We can think of this setting as a contract between an insurer and an insuree, where the insuree is partly unaware and effort affects the probability of incurring a loss. The additional participation effect gives the insurer incentives to reveal severe calamities to the insuree, such that the insuree is willing to buy insurance at a higher price. 7.5. Unawareness and zero probability beliefs A frequently raised concern is whether unawareness is observationally equivalent to full awareness with zero probability beliefs (Li, 2008). Epistemically, unawareness has very different properties from zero probability beliefs. An agent is unaware if and only if he assigns probability zero to an event and to its negation (Heifetz et al., 2013). Schipper (2013) shows how this feature also implies behavioral differences between unawareness and zero probability beliefs. However, the results derived in my model can be generated in a framework with full awareness and zero probability beliefs. Under the interpretation of heterogeneous priors, there are some caveats to be taken into account. In order to derive my results in a framework with full awareness and zero probability beliefs, the agent needs to update a zero probability prior to a non-zero posterior. Note that such updating cannot be interpreted as a consequence of the arrival of new information, since information in the standard state space model expands the set of null states instead of narrowing it. Generating my results in the standard state space framework requires a model that allows for manipulation of beliefs rather than revelation of information. Allowing the principal to manipulate the agent’s beliefs without the presence of hard information is rather difficult to motivate. Further, it is important to note that reporting the true distribution is not incentive compatible for the principal if lying is possible. Consequently, there are strong assumptions on the message set available to the principal necessary.28 Given these caveats, asymmetric awareness seems to be a more natural way to think about this environment and the arising tradeoff. 8. Conclusion This paper incorporates asymmetric awareness in the classical principal–agent model. It shows that the principal makes the agent strategically aware and that the optimal contract can be incomplete. Enlarging the agent’s awareness involves a tradeoff between participation and incentives. The cost of disclosing contingencies to the agent is the payment in the states that the agent is initially unaware of. The gain of disclosing contingencies to the agent is the richer information structure that is used to induce incentives. Hence, it is profitable to announce contingencies that have a low probability but are highly correlated with the effort of the agent. If we allow for competition among principals, there exists a symmetric Nash equilibrium in which the agent is fully aware and principals make zero profits. Remarkably, there may exist other equilibria in which the agent stays unaware, even when competition is tight. The existence of incomplete contracts in equilibrium may have important implications for welfare because whenever the agent is left unaware, the principal uses an inefficient information structure to induce incentives. In the proposed model, the principal is able to implement zero payments whenever there is an event the agent is initially unaware of. This may not be feasible in real-life contracting situations. If, for example, the compensation scheme is restricted to be monotone in outcome, it is most costly to disclose low outcomes to the contracting partner. Hence, the magnitude of

27 Due to the limited liability constraint the principal would like to scale up both P and C . In order to have a solution, one has to assume that such a contract elicits suspicion on the side of the agent. 28 One may argue that also in the framework with unawareness an implicit assumption is that the principal can only reveal “true” contingencies. One could imagine a case, where the principal can include virtual events, but since I assume that the agent, once aware, completely understands all consequences and the probability distribution of a contingency, the restriction to “true” contingencies seems to be natural.

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the participation effect depends on the revealed outcome. Restricting the set of feasible contracts adds interesting features to the optimal compensation scheme and revelation strategy, but as long as the contracting partner with superior awareness is able to profit from the other’s limited understanding of the underlying uncertainties the basic tradeoff prevails. An open question is how the results of the basic model change in a repeated game setting. Whenever a contingency outside the agent’s awareness is realized, the agent observes an outcome considered impossible and consequently becomes aware of his initial unawareness. This raises the question how such a discovery affects the agent’s updated understanding of the world, i.e. whether the agent passively updates his state space or whether he understands that there may be other events he is unaware of. To answer these questions, it is necessary to enter the debate on decision making under awareness of unawareness, a largely unexplored field in the literature. It seems plausible that also in repeated games asymmetric awareness has interesting implications and thus poses interesting challenges for future research. Acknowledgments I am very grateful to my supervisor Piero Gottardi. I would also like to thank Arpad Abraham, Wouter Dessein, Erik Eyster, Leonardo Felli, Paul Heidhues, Aviad Heifetz, Martin Meier, Salvatore Modica, Stephen Morris, In-Uck Park, Nicola Pavoni, Michele Piccione, Andrew Postlewaite, Andrea Prat, Luis Santos-Pinto, Balazs Szentes, Jean-Marc Tallon and two anonymous referees for very helpful comments. Appendix A A.1. Theoretical foundations of the unawareness structure My model adopts the unawareness structure introduced by Heifetz et al. (2006). They propose a generalized state space model that allows for non-trivial unawareness in multi-agent settings and strong properties of knowledge. In Heifetz et al. (2008), the authors provide complete and sound axiomatization for their class of unawareness structures. For ease of exposition, the basic set up in this paper is a strongly simplified version of the original model, foregoing the formal introduction of state spaces, projections among them, events, etc. This section does not present the generalized state space model in detail, but provides some insights on how my basic model is built on the foundational literature. The unawareness structure proposed by Heifetz et al. (2006/2013) consists of a lattice of disjoint space spaces S = { S Θ }Θ⊆Θ , with a partial order  on S . S Θ   S Θ  means that S Θ  is more expressive than S Θ  , so the spaces are ordered  according to their richness  in terms of facts that they can describe. The upmost state space S Θ is interpreted as the objective SΘ state space. Let Ω := Θ⊆Θ  denote the union of all state spaces with typical element ω . Any ω ∈ S Θ  can be interpreted 

, the starting point of the basic model. as a vector of realizations of the random variables in Θ S 

S 

For any S Θ  , S Θ  ∈ S such that S Θ   S Θ  , there is a surjective projection r S Θ : S Θ  → S Θ  , where r S Θ (ω) is the restricted description of

Θ

Θ

ω ∈ S Θ  in the limited vocabulary of S Θ  . Let g ( S Θ  ) = { S Θ  ∈ S : S Θ   S Θ  } denote the set of

state spaces that are at least as expressive as S Θ  . Further, given a set of states D ⊆ S Θ  , let D ↑ =



S Θ  −1 (D) S Θ  ∈ g ( S Θ  ) (r S Θ  )

denote all ω ∈ Ω that describe D in at least as expressive vocabulary as S Θ  . Then an event is a pair ( D ↑ , S Θ  ) with D ⊆ S Θ  and S Θ  ∈ S . Back to the basic model an elementary event that some proposition θ ∈ Θ is true is denoted by (ω↑ , S θ ), where ω is the state in S θ in which θ is true (note that S θ has two elements). Consequently, ω↑ is the set of states in Ω where the proposition θ is expressible and true, ( S θ \ω)↑ is the set of states in Ω where proposition θ is expressible and false and Ω\{ω↑ ∪ ( S \ω)↑ } is the set of states where neither the event of θ being true nor false is  and θ ∈  than S Θ ⊂ Ω\{ω↑ ∪ ( S \ω)↑ }. /Θ expressible. If the agent is aware of Θ Similar to the basic model, define y = f (ω), ω ∈ S Θ , where f is a one-to-one function. The agent cannot express S Θ and consequently cannot know function f . It is assumed that the agent’s perceived outcome is equal to the actual outcome when no unforeseen contingency is realized. Let (ωθ=0 ↑ , S θ ), ωθ=0 ∈ S θ denote the event that θ = 0. Then the agent’s perceived outcome function is defined by

 f (ω) := f





S

r S Θ

− 1

Θ

 (ω) ∩

 

(ωθ =0 )↑

 ,

ω ∈ S Θ .

 θ ∈Θ\Θ

It is easy to check that this coincides with the original specification of the outcome function. In Heifetz et al. (2013) the generalized state space model is augmented by probabilistic beliefs. Let  of μ on S Θ is defined by measure on S Θ . Then the marginal μ



(ω) := μ r SS Θ μ Θ

− 1

 (ω) ,

μ denote a probability

ω ∈ S Θ .

(ω|e ), ω ∈ S Θ In the basic set up we are interested in conditional probabilities on effort, so define μ(ω|e ), ω ∈ S Θ and μ ( y ) = μ ( analogously. Then let π ( y |e ) := μ( f −1 ( y )|e ) and π f −1 ( y )|e ). Under the assumption of independence, this yields  e ) π ( y |e) = Π(Θ| π ( y |e), the original definition in the model.

S. Auster / Games and Economic Behavior 82 (2013) 503–521

519

A.2. Proofs A.2.1. Proof of Proposition 4.1 ∗ . The optimal compensation scheme is  Suppose Θ A ⊂ Θ C ∗ ( y) = . Then the expected payment is y ∈ Y \Y





Pr θ = 0|e H

∗ θ ∈Θ\Θ

which is greater than optimal.







 and  υ −1 (U¯ + e H ) for all y ∈ Y C ∗ ( y ) = 0 for all



υ −1 U¯ + e H ,

θ∈Θ\Θ A Pr[θ

∗ cannot be = 0|e H ]υ −1 (U¯ + e H ), due to the assumption π ( y |e ) > 0, ∀ y ∈ Y . Hence Θ

A.2.2. Proof of Lemma 5.1    such that ( y |e H ) = y∈Y π ( y |e L ) = 1 and π (.|e H ) = π (.|e L ) there must exist some y ∈ Y Suppose λ = 0. Since π y ∈Y 1 H L  ( y |e ) − π ( y |e ) < 0. But since γ  0, λ = 0 would imply that υ  (C ( y))  0 for some y ∈ Y , which violates the assumption π

υ  (·) > 0. Hence λ > 0. Now suppose γ = 0. Then, the first-order conditions of the optimization problem imply that compensation is fixed across outcomes within the agent’s awareness. But this implies that the incentive constraint is no longer satisfied. Hence, γ > 0.

A.2.3. Proof of Lemma 5.2 Let S˜ denote the state space, let y˜ denote the output function with range Y˜ and let π˜ (.|e ) denote the probability belief  be the mapping from set Y˜ to set Y , where ρ ( y˜ (˜s)) =   ∪ Z . Further, let ρ : Y˜ −→ Y given awareness Θ y ( s ) and  s ∈ S is a ˜ Now consider the compensation scheme C˜ with C˜ ( y ) = C C (ρ ( y )) for all y ∈ Y˜ . Note that C˜ satisfies subsequence of s˜ ∈ S.  Θ both participation and incentive constraint with equality. Suppose Pr[θ = 1|e H ] = Pr[θ = 1|e L ], ∀θ ∈ Z . Then, for any y ∈ Y˜ we have

( y |e L ) π˜ ( y |e L ) π = . ( y |e H ) π˜ ( y |e H ) π Hence, C C satisfies the first-order conditions and consequently solves the optimization problem. C Z = 0. Θ Θ  such that Now, suppose Pr[θ = 1|e H ] = Pr[θ = 1|e L ] for some θ ∈ Z . Then, there must exist some y , y  ∈ ρ −1 ( y ), y ∈ Y

π˜ ( y |e L ) π˜ ( y  |e L ) =

. π˜ ( y |e H ) π˜ ( y  |e H ) Consequently, C C does not satisfy the first-order conditions. Hence, C C is feasible but not optimal, which implies that E [C C

Θ

Θ∪ Z

Θ

C Z H (Y |e H )] < E [C Θ  (Y |e )] and C Θ  > 0.

A.2.4. Proof of Lemma 5.3 θ = E [C C (Y )|e H ] − E [C C H θ To show that C Θ ΘA Θ A ∪θ (Y )|e ] is monotonically increasing in it is sufficient to show that A

C E [C Θ (Y )|e H ] is monotonically decreasing in θ . W.l.o.g. assume Pr[θ = 1|e H ] > Pr[θ = 1|e L ]. Let E Θ A ∪θ denote the exA ∪θ pectation operator with respect to awareness Θ A ∪ θ . Then we have:

E Θ A ∪θ













C C υ CΘ (Y ) |e , θ = 1 > E Θ A ∪θ υ C Θ (Y ) |e , θ = 0 , e = e L , e H , A ∪θ A ∪θ

which follows directly from the first-order conditions. The incentive constraint can be rewritten as





Pr θ = 1|e H E Θ A ∪θ

















C C υ CΘ (Y ) |e H , θ = 1 + Pr θ = 0|e H E Θ A ∪θ υ C Θ (Y ) |e H , θ = 0 − e H A ∪θ A ∪θ



 C



 C

  = Pr θ = 1|e L E Θ A ∪θ υ C Θ (Y ) |e L , θ = 1 + Pr θ = 0|e L E Θ A ∪θ υ C Θ (Y ) |e L , θ = 0 − e L . A ∪θ A ∪θ

C Now, consider probability Pr[θ = 1|e L ] − ε with ε > 0. Under C Θ and Pr[θ = 1|e L ] − ε the participation constraint is A ∪θ clearly satisfied with equality. The incentive constraint implies





Pr θ = 1|e H E Θ A ∪θ

















C C υ CΘ (Y ) |e H , θ = 1 + Pr θ = 0|e H E Θ A ∪θ υ C Θ (Y ) |e H , θ = 0 − e H A ∪θ A ∪θ



 C



 C

  > Pr θ = 1|e L E Θ A ∪θ υ C Θ (Y ) |e L , θ = 1 + Pr θ = 0|e L E Θ A ∪θ υ C Θ (Y ) |e L , θ = 0 − e L A ∪θ A ∪θ   C

 C

   − ε E Θ A ∪θ υ C Θ (Y ) |e L , θ = 1 − E Θ A ∪θ υ C Θ (Y ) |e L , θ = 0 . A ∪θ A ∪θ

C Under the optimal transfer rule both constraints are satisfied with equality. Consequently, given Pr[θ = 1|e L ] − ε , C Θ is A ∪θ H feasible but not optimal. The same line of reasoning applies to Pr[θ = 1|e ], in which case both constraints are slack. Thus, the expected payment given information structure Θ A ∪ θ is decreasing in θ .

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A.2.5. Proof of Proposition 5.5 H If E Θ ∗ [ Y − C Θ ∗ (Y )|e ] < 0, the agent thinks that the principal would be strictly better off by not offering the contract. H L ¯ the agent cannot rationalize why the principal proposes an incentive Similarly if E Θ ∗ [ Y − C Θ ∗ (Y )|e ] < E Θ ∗ [ Y |e ] − C ∗ ∗   compatible contract. Hence, (Θ , C ) cannot be justifiable. H H L ¯. Now suppose E Θ ∗ [ Y − C Θ ∗ (Y )|e ]  0 and E Θ ∗ [ Y − C Θ ∗ (Y )|e ]  E Θ ∗ [ Y |e ] − C ∗ = Θ , principal and agent share the same beliefs. Hence,  C ∗ : If Θ C ∗ maximizes the principal’s expected Justifiability of  C ∗ ∗    payoff from the agent’s perspective. If Θ = Θ , C ( y ) = C ∗ ( y ) for all y ∈ Y . Since the agent thinks that the contract is Θ

C ∗ maximizes the principal’s payoff according to the beliefs of the agent. complete,  ∗ : Θ ∗ is optimal for the principal given the agent’s beliefs if Justifiability of Θ





C H E CΘ ∗ Y |e







∗ \ Z θ ∈Θ







C H Pr θ = 0|e H E C Θ ∗ \ Z (Y )|e ,

∗ \Θ A . This coincides with the optimality condition of the principal. Hence, whenever E Θ∗ [Y − C Θ∗ (Y )|e H ]  0 for any Z ⊆ Θ H L ¯ , (Θ ∗ ,  and E Θ [ Y − CΘ C ∗ ) is justifiable. ∗ ∗ (Y )|e ]  E Θ ∗ [ Y |e ] − C A.2.6. Proof of Proposition 6.1 In the proposed equilibrium principals make zero profits and the agent obtains the positive second-best surplus. Consider j = Θ and C j = C ∗ for all j = 1, . . . , i − 1, i + 1, . . . , N. A deviation in Θ i leaves a deviation of principal i. The strategies are Θ 1 ∪ · · · ∪ Θ i ⊆ Θ . A deviation in C i is not N = Θ for all Θ the expected payoff unaffected because the agent is aware of Θ profitable for the standard Bertrand argument. C ∗ maximizes the agent’s expected utility subject to the zero profit constraint and the incentive constraint. A deviation C i = C ∗ such that the incentive constraint is satisfied must make either the agent or the principal worse off. If the agent is worse off, he rejects the contract and the expected payoff is zero. If the principal is worse off, he has a negative expected payoff. A deviation in C i = C ∗ such that the incentive constraint is not satisfied makes either the agent worse off or the principal worse off or both by Assumption 3. Hence, there is no profitable deviation and (Θ, C ∗ ) is an equilibrium. A.2.7. Proof of Proposition 7.1 When e L is optimal, the principal solves

max

 C (·) Θ,

 





π y |e L y − C ( y )

y ∈Y

subject to

      y |e L υ C ( y ) − e L  U , π  y ∈Y

C ( y )  0,

∀y ∈ Y.

 and C L ( y ) = 0, ∀ y ∈ Y \Y . The expected  is C L ( y ) = υ −1 (U¯ − e L ), ∀ y ∈ Y The optimal compensation scheme for a given Θ payment is





E C L (Y ) =





Pr θ = 0|e L







υ −1 U¯ − e L ,

 θ ∈Θ\Θ

 = ΘA . which is minimized if Θ References Benabou, R., Tirole, J., 2003. Intrinsic and extrinsic motivation. Rev. Econ. Stud. 70 (3), 489–520. Board, O.J., Chung, K.S., 2011. Object-based unawareness: Theory and applications. Mimeo. Chen, Y.J., Zhao, X., 2009. Contractual traps. In: Proceedings of the 12th Conference on Theoretical Aspects of Rationality and Knowledge, pp. 51–60. De la Rosa, L.E., 2011. Overconfidence and moral hazard. Games Econ. Behav. 73 (2), 429–451. Dekel, E., Lipman, B., Rustichini, A., 1998. Standard state-space models preclude unawareness. Econometrica 66 (1), 159–173. Filiz-Ozbay, E., 2012. Incorporating unawareness into contract theory. Games Econ. Behav. 76 (1), 181–194. Gabaix, X., Laibson, D., 2006. Shrouded attributes, consumer myopia, and information suppression in competitive markets. Quart. J. Econ. 121 (2), 505–540. Galanis, S., 2013. Unawareness of theorems. Econ. Theory 52 (1), 41–73. Grossman, S., Hart, O., 1983. An analysis of the principal–agent problem. Econometrica 51 (1), 7–45. Heifetz, A., Meier, M., Schipper, B.C., 2006. Interactive unawareness. J. Econ. Theory 130 (1), 78–94. Heifetz, A., Meier, M., Schipper, B.C., 2008. A canonical model for interactive unawareness. Games Econ. Behav. 62 (1), 304–324. Heifetz, A., Meier, M., Schipper, B.C., 2013. Unawareness, beliefs and speculative trade. Games Econ. Behav. 77 (1), 100–121. Holmström, B., 1979. Moral hazard and observability. Bell J. Econ. 10 (1), 74–91. Karni, E., Vierø, M.-L., 2012. Reverse bayesianism: A choice-based theory of growing awareness. Mimeo. Li, J., 2008. A note on unawareness and zero probability. Mimeo. Li, J., 2009. Information structures with unawareness. J. Econ. Theory 144 (3), 977–993.

S. Auster / Games and Economic Behavior 82 (2013) 503–521

Modica, S., Rustichini, A., 1994. Awareness and partitional information structures. Theory Dec. 37 (1), 107–124. Modica, S., Rustichini, A., 1999. Unawareness and partitional information structures. Games Econ. Behav. 27 (2), 265–298. Modica, S., Rustichini, A., Tallon, J.M., 1998. Unawareness and bankruptcy: A general equilibrium model. Econ. Theory 12 (2), 259–292. Ozbay, E., 2008. Unawareness and strategic announcements in games with uncertainty. Mimeo. Santos-Pinto, L., 2008. Positive self-image and incentives in organizations. Econ. J. 118 (531), 1315–1332. Schipper, B.C., 2013. Awareness-dependent subjective expected utility. Int. J. Game Theory 42 (3), 725–753. Von Thadden, E.L., Zhao, X.J., 2012a. Incentives for unaware agents. Rev. Econ. Stud. 79 (3), 1151–1174. Von Thadden, E.L., Zhao, X.J., 2012b. Multi-task agency with unawareness. Mimeo. Zhao, X.J., 2008. Moral hazard with unawareness. Ration. Soc. 20 (4), 471–496.

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